We introduce the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of $0$-, $1$- and $2$-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by counting colourings of the $1$-cells with elements of a finite group $G$, subject to a “flatness” condition for each $2$-cell. These invariants are also described in a TQFT setting, which is not the same as the usual $2$-dimensional TQFT framework. We study the properties of functions which arise in this context,associated to the disk, the cylinder and the pants surface, and derive general properties of these functions from topology. One such property states that the number of conjugacy classes of $G$ equals the commuting fraction of $G$ times the order of $G$.

We will comment on the extension of these invariants to 2-groups and their (higher) gauge theory interpretation.

This is work done in collaboration with Diogo Bragança (Dept. Physics, IST).